Bayesian vs Frequentist Comparison Cheat Sheet

Bayesian and frequentist statistics are two major ways to use data to make conclusions under uncertainty. This cheat sheet compares how each approach treats probability, parameters, evidence, and conclusions. Students need it because the same data can lead to different wording and reasoning depending on the statistical framework.

Understanding the comparison helps students interpret studies, experiments, polls, and statistical claims more carefully.

Frequentist methods treat parameters as fixed but unknown values and use repeated sampling to describe uncertainty. Bayesian methods treat parameters as uncertain and update beliefs using Bayes' theorem, $P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}$ . Key comparisons include confidence intervals versus credible intervals, $p$ -values versus posterior probabilities, and likelihood-based evidence versus prior-updated evidence.

Both approaches use data, models, and assumptions, but they answer different statistical questions.

Key Facts

Bayes' theorem is $P(H\mid D)=\frac{P(D\mid H)P(H)}{P(D)}$ , where $H$ is a hypothesis and $D$ is the observed data.
In frequentist statistics, an unknown parameter such as $\theta$ is fixed, and uncertainty comes from the random sample.
In Bayesian statistics, an unknown parameter such as $\theta$ is treated as uncertain, and a probability distribution describes beliefs about its possible values.
A frequentist confidence interval has the form $\hat{\theta}\pm z^{\ast}\cdot SE$ , where $\hat{\theta}$ is an estimate and $SE$ is its standard error.
A Bayesian posterior distribution is proportional to likelihood times prior, written as $P(\theta\mid D)\propto P(D\mid \theta)P(\theta)$ .
A $95\%$ confidence interval means that, over many repeated samples, about $95\%$ of intervals made by the same method would contain the true parameter.
A $95\%$ credible interval means that, given the model, data, and prior, there is posterior probability $0.95$ that $\theta$ lies in the interval.
A frequentist $p$ -value is $P(\text{data at least as extreme as observed}\mid H_0)$ , not the probability that $H_0$ is true.

Vocabulary

Prior distribution: A probability distribution, written $P(\theta)$ , that represents beliefs about a parameter before observing the current data.
Likelihood: The likelihood, written $P(D\mid \theta)$ , measures how compatible the observed data $D$ are with different parameter values $\theta$ .
Posterior distribution: The posterior distribution, written $P(\theta\mid D)$ , combines the prior and likelihood to describe updated beliefs after seeing the data.
Confidence interval: A confidence interval is a frequentist interval estimate built by a method that captures the true parameter in a stated proportion of repeated samples.
Credible interval: A credible interval is a Bayesian interval that contains a stated amount of posterior probability for a parameter.
P-value: A $p$ -value is the probability, assuming $H_0$ is true, of getting results at least as extreme as the observed results.

Common Mistakes to Avoid

Saying a $95\%$ confidence interval has a $95\%$ chance of containing the true parameter is wrong because, in frequentist statistics, the parameter is fixed and the interval is random.
Treating a $p$ -value as $P(H_0\mid D)$ is wrong because a $p$ -value is calculated as $P(\text{data at least as extreme as observed}\mid H_0)$ .
Ignoring the prior in a Bayesian analysis is wrong because the posterior $P(\theta\mid D)$ depends on both the prior $P(\theta)$ and the likelihood $P(D\mid \theta)$ .
Comparing a credible interval and a confidence interval as if they mean the same thing is wrong because they answer different questions about uncertainty.
Assuming Bayesian methods are always subjective and frequentist methods are always objective is wrong because both approaches require modeling choices and assumptions.

Practice Questions

1 A diagnostic test has $P(+\mid \text{disease})=0.98$ , $P(+\mid \text{no disease})=0.05$ , and $P(\text{disease})=0.02$ . Use Bayes' theorem to find $P(\text{disease}\mid +)$ .
2 A sample has $\hat{p}=0.62$ and $SE=0.04$ . Using $z^{\ast}=1.96$ , compute the approximate $95\%$ confidence interval $\hat{p}\pm z^{\ast}\cdot SE$ .
3 A Bayesian analysis gives a posterior distribution with a $95\%$ credible interval of $[0.48,0.72]$ for $\theta$ . Write one correct sentence interpreting this interval.
4 A study reports a $p$ -value of $0.03$ . Explain what this means in frequentist terms and why it does not prove that the alternative hypothesis is true.

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Bayesian vs Frequentist Comparison Cheat Sheet

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